Measurements of High-Frequency Sound Propagation in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="normal">He</mml:mi></mml:mrow><mml:mprescripts /><mml:mrow /><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow /><mml:mrow /></mml:mmultiscripts></mml:mrow></mml:math>-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>B</mml:mi></mml:math>

Mast, David; Sarma, Bimal K.; Owers‐Bradley, J. R.; Calder, I. D.; Ketterson, J. B.; Halperin, W. P.

doi:10.1103/physrevlett.45.266

Cited by 119 publications

(30 citation statements)

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“…7 ' 8 At a given pressure, v g vs a> can be fitted by least squares to yield o) sq and A sq . Our results are a; sq /A BCS (0) = 1.501 ±0.003 and A sq (0) = 0.010 (4) ±0.001 at a pressure of 13.1 bars. A BCS (T) is taken from numerical calculations.…”

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“…4 In order to take data we cooled the 3 He at a pressure in the interval 11.7 to 15.5 bars to a minimum temperature T/T c ^ 0.2 and allowed it to slowly warm up through the superfluid over a period of several days. The received acoustic pulses at frequencies of 12.06, 36.33, 60.56, 84.83, 109.10, and 133.33 MHz were signal averaged and the peaks were fitted to Gaussians.…”

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Group-Velocity Spectroscopy of Collective Modes in SuperfluidHe3-B

et al. 1980

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The group velocity of high-frequency sound for T «T C , and the group velocity and attenuation of sound at higher temperatures, have been measured and analyzed to give the frequencies and coupling strengths of two order-parameter collective modes in 3 He-fi. For the squashing mode at 13.1 bars and T-*-(), a; sq /A B cs = l o 501 and the coupling constant A sq = 0.010± 0.001. The recently discovered collective mode at a; y /A BCS =1.075 (T -*0) has a coupling constant determined from the group velocity to be A. y = 5.8X10" 6 o PACS numbers: 67.50.Fi Superfluid Fermi systems are characterized in part by an energy gap A(T) at the Fermi surface arising from the creation of energy states occupied by Cooper pairs. The characteristic frequency of the system, A(0) = 1.164k B T c /h (in the BCS theory), is in the microwave regime for superconductors but varies between 40 and 100 MHz for superfluid 3 He, a frequency range easily accessible by standard ultrasonic techniques. In 3 He the order parameter has an internal structure which may undergo oscillatory distortions. These collective modes 1 ' 2 occur at frequencies that scale approximately with the gap and are characteristic of the pairing state.In superfluid 3 Be-B two collective modes have been found experimentally, the squashing mode, 3 o) sq , and a new mode, 4 ' 5 w v , that has not yet been identified. Previously it has been difficult to obtain detailed information about the temperature dependence of the squashing-mode frequency outside of a narrow interval within ~1% of T c ; at lower temperatures the mode produces an enormous sound attenuation. In this work we report the application of a high-resolution spectroscopic tool, group-velocity spectroscopy (GVS). This technique is particularly appropriate for the investigation of strong anomalies such as the squashing mode. The essence of this method is that, at low temperatures where quasiparticle collision broadening is small, the group velocity has a nonzero spectral width determined by the coupling strength of the sound wave to the order-parameter mode. This spectrum can be easily measured. On the other hand, the attenuation anomaly is almost a 6 function superimposed on the other strong attenuation mechanism, pair breaking.It is useful to have a phenomenological descrip-tion of sound propagation near a collective-mode resonance. 6 We represent the distortion of the order parameter produced by some external field by a scalar amplitude 6. In the absence of external fields we expect 6 to return to an equilibrium value of zero in an oscillatory manner with frequency components w^^AfT) and time constant T. Each collective mode j is distinguished by a quantity a i which can be expected to be almost independent of pressure and temperature. Sound waves couple to the order parameter, producing forced oscillation in 6 at the applied frequency co. In order that 6 be zero in the absence of density changes and that its oscillations be undamped in the absence of quasiparticle collisions ( T -*oo) ? § must couple only to second...

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Group-Velocity Spectroscopy of Collective Modes in SuperfluidHe3-B

et al. 1980

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“…1 we plot^2 + (T) for the two extreme cases jF 2 a =0, x 3~1 = -0A32 (hereafter case I) and F 2 a = -1.56, j^" 1 = 0 (hereafter case II) corresponding to co 2+ (0) = 1.075A BCS (0), with A BCS (T) the weak-coupling BCS energy gap. 6 For comparison we also show the "bare"g factors calculated for F 2 3 ' a = x 3 -1 = 0. If we take A(T) = A BCS( T ), then f°r f ix ed values of F Q a , F 2 s * a , and x 3 , g 2± and ct> L are universal functions of T/T c , as are to 2± (T)/A(T) and dco 2± /dT.…”

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“…The Zeeman splitting of the 2 -mode has yet to be observed, but the splitting of the 2 + mode has been measured by Avenel, Varoquaux, and Ebisawa; 4 their observation of a fivefold splitting of the new ultrasonic absorption line discovered by Giannetta et al 5 and Mast et al 6 established conclusively that this absorption line is associated with the 2+ mode. Equation (2) gives 6oo(T,H), the linear splitting of the modes at a fixed temperature, but experimentally one measures 6T(w, H), the splitting in temperature of the modes at a fixed frequency.…”

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Interaction Effects on the Zeeman Splitting of Collective Modes in SuperfluidHe3-B

Sauls

Serene

1982

Phys. Rev. Lett.

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Exact calculations of the B-phase collective-mode frequencies in a magnetic field are presented, and they provide new information about the effective mass, the pairing interaction, and the Landau parameters. The calculation of the Zeeman splitting for the 2+ mode (real squashing mode) is consistent with the zero-field frequency and agrees quantitatively with the measurements of Avenel et al. 9 for a range of values of m* which includes some, but not all, of the differing experimental results for this important parameter.PACS numbers: 67.50.Fi Superfluid 3 He supports a number of orderparameter collective modes with temperaturedependent frequencies of the formwhere a and b are weakly temperature-dependent functions of order 1. The zero-sound dispersion relation co=c Q q crosses the dispersion relations for these modes when qv ¥ /A(T)~v F /c 0 «l, and hence in discussing their excitation by ultrasound one can generally neglect the dispersion of the collective modes. It is suggestive to think of the collective modes as excited states of the Cooper pairs, and to regard their excitation by zero sound as an ultrasonic Cooper-pair spectroscopy analogous to the optical spectroscopy of diatomic molecules. This analogy is strengthened by studies of zero-sound absorption in a magnetic field. For example, Schopohl and Tewordt 1 * 2 predicted that a small magnetic field should produce a linear "Zeeman splitting" of both the real and imaginary J=2 order-parameter modes in 3 He-i3 (the 2 + and 2 -modes, in our previous notation, 3 which we follow henceforth). The collective modes are then labeled by ffi j, their angular momentum projection along H, and their frequencies have the form u> 2± (m^( T, H) = u 2 JL7 1 ) + m J g 2 Ji7 i )w L {T, H\where oo L (T,H), the temperature-dependent effective Larmor frequency, is proportional to the magnetic field. The Zeeman splitting of the 2 -mode has yet to be observed, but the splitting of the 2 + mode has been measured by Avenel, Varoquaux, and Ebisawa; 4 their observation of a fivefold splitting of the new ultrasonic absorption line discovered by Giannetta et al. 5 and Mast et al. 6 established conclusively that this absorption line is associated with the 2+ mode. Equation (2) gives 6oo(T,H), the linear splitting of the modes at a fixed temperature, but experimentally one measures 6T(w, H), the splitting in temperature of the modes at a fixed frequency. If we expand Eq. (2) around T 0 , the temperature of the absorption line in zero field, we obtain oT ((l3Schopohl and Tewordt 7 calculated the Zeeman splittings of the 2 ± modes keeping only the Landau parameter F 0 a and assuming a pure 1 = 1 pairing interaction. With the help of a new identity given below [Eq. (14)], their results can be written as co L (T,H) = yH/[l + F 0^ + iY)].

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“…Indeed ultra-sound absorption spectroscopy provided the first detection of the Higgs mode in a BCS condensate. 59,60 The definitive identification of the absorption resonance as the J c = 2 + Higgs mode was made by Avenel et al who observed the five-fold Zeeman splitting of the zero-sound absorption resonance in an applied magnetic field. 61 Acoustic spectroscopy provides precision measurements of the mass of the J c = of the masses of the J c = 2 ± modes using longitudinal and transverse sound spectroscopy, combined with measurements of the velocities of zero-sound, first-sound, and the magnetic susceptibilties in both the normal-and superfluid phases of 3 He.…”

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On the Nambu fermion-boson relations for superfluidHe3

Sauls

Mizushima

2017

Phys. Rev. B

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Superfluid 3 He is a spin-triplet (S = 1), p-wave (L = 1) BCS condensate of Cooper pairs with total angular momentum J = 0 in the ground state. In addition to the breaking of U(1) gauge symmetry, separate spin or orbital rotation symmetry is broken to the maximal sub-group, SO(3) S × SO(3) L → SO(3) J . The Fermions acquire mass, m F ≡ ∆, where ∆ is the BCS gap. There are also 18 Bosonic excitations -4 Nambu-Goldstone (NG) modes and 14 massive amplitude Higgs (AH) modes. The Bosonic modes are labeled by the total angular momentum, J ∈ {0, 1, 2}, and parity under particle-hole symmetry, c = ±1. For each pair of angular momentum quantum numbers, J, J z , there are two Bosonic partners with c = ±1. Based this spectrum Nambu proposed a sum rule connecting the Fermion and Boson masses for BCS type theories, which for 3 He-B is M 2 J, + + M 2 J, − = 4m 2 F for each family of Bosonic modes labeled by J, where M J, c is the mass of the Bosonic mode with quantum numbers (J, c). Nambu's sum rule (NSR) has recently been discussed in the context of Nambu-Jona-Lasinio models for physics beyond the standard model to speculate on possible partners to the recently discovered Higgs Boson at higher energies. Here we point out that Nambu's Fermion-Boson mass relations are not exact. Corrections to the Bosonic masses from (i) leading order strong-coupling corrections to BCS theory, and (ii) polarization of the parent Fermionic vacuum lead to violations of the sum-rule. Results for these mass corrections are given in both the T → 0 and T → T c limits. We also discuss experimental results, and theoretical analysis, for the masses of the J c = 2 ± Higgs modes and the magnitude of the violation of the NSR. I. INTRODUCTIONOne of the key features of spontaneous symmetry breaking in condensed matter and quantum field theory is the emergence of new elementary quanta -phonons in crystalline solids, magnons in ferromagnets, the Higgs and gauge bosons of the standard model. In the latter example, spontaneous symmetry breaking (SSB) in the BCS theory of superconductivity played an important role in theoretical models for the mass spectrum of elementary particles. [1][2][3] In BCS superfluids the binding of Fermions into Cooper pairs leads to an energy gap, ∆, in the Fermion spectrum, i.e. Fermions in the broken symmetry phase (Bogoliubov quasiparticles) acquire a mass m F = ∆, while condensation of Cooper pairs leads to the breaking of global U(1) gauge symmetry, the generator being particle number. The latter also implies that the Bogoliubov Fermions are no longer particle number (Fermion "charge") eigenstates, but coherent superpositions of normal-state particles and holes. Charge conservation is ensured by an additional contribution to the charge current -a collective mode of the broken symmetry phase. This massless Bosonic excitation of the phase of condensate amplitude 4,5 is the NambuGoldstone (NG) mode associated with broken U(1) symmetry, and is manifest as a phonon in neutral superfluid 3 He. II. NAMBU'S MASS RELATIONSNambu and Jona...

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Measurements of High-Frequency Sound Propagation inHe3-B

Cited by 119 publications

References 8 publications

Group-Velocity Spectroscopy of Collective Modes in SuperfluidHe3-B

Group-Velocity Spectroscopy of Collective Modes in SuperfluidHe3-B

Interaction Effects on the Zeeman Splitting of Collective Modes in SuperfluidHe3-B

On the Nambu fermion-boson relations for superfluidHe3

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